Acceleration is a familiar word. For non-engineers, it most often comes across in news articles and releases. Acceleration of development, cooperation, and other social processes. The original meaning of this word is associated with physical phenomena. How to find the acceleration of a moving body, or acceleration, as an indicator of the power of a car? Could it have other meanings?
What happens between 0 and 100 (term definition)
An indicator of a car's power is considered to be the time it takes to accelerate from zero to hundreds. What happens in between? Let's look at our Lada Vesta with its stated 11 seconds.
One of the formulas for finding acceleration is written like this:
a = (V 2 - V 1) / t
In our case:
a - acceleration, m/s∙s
V1 - initial speed, m/s;
V2 - final speed, m/s;
Let's bring the data into the SI system, namely, km/h will be converted to m/s:
100 km/h = 100,000 m / 3600 s = 27.28 m/s.
Now you can find the acceleration of the "Kalina":
a = (27.28 - 0) / 11 = 2.53 m/s∙s
What do these numbers mean? An acceleration of 2.53 meters per second per second means that for every second the speed of the “car” increases by 2.53 m/s.
When starting from a place (from scratch):
- in the first second the car will accelerate to a speed of 2.53 m/s;
- for the second - up to 5.06 m/s;
- by the end of the third second the speed will be 7.59 m/s, etc.
Thus, we can summarize: acceleration is the increase in the speed of a point per unit time.
Newton's second law, it's not difficult
So, the acceleration value has been calculated. It's time to ask where this acceleration comes from, what is its primary source. There is only one answer - strength. It is the force with which the wheels push the car forward that causes its acceleration. And how to find acceleration if the magnitude of this force is known? The relationship between these two quantities and the mass of a material point was established by Isaac Newton (this did not happen on the day when an apple fell on his head, then he discovered another physical law).
And this law is written like this:
F = m ∙ a, where
F - force, N;
m - mass, kg;
a - acceleration, m/s∙s.
In relation to a product of the Russian automobile industry, it is possible to calculate the force with which the wheels push the car forward.
F = m ∙ a = 1585 kg ∙ 2.53 m/s∙s = 4010 N
or 4010 / 9.8 = 409 kg∙s
This means that if you do not release the gas pedal, the car will accelerate until it reaches the speed of sound? Of course not. Already when it reaches a speed of 70 km/h (19.44 m/s), the frontal air resistance reaches 2000 N.
How to find the acceleration at the moment when the Lada “flies” at such a speed?
a = F / m = (F wheels - F resistance) / m = (4010 - 2000) / 1585 = 1.27 m/s∙s
As you can see, the formula allows you to find both acceleration, knowing the force with which the engines act on the mechanism (other forces: wind, water flow, weight, etc.), and vice versa.
Why is it necessary to know acceleration?
First of all, in order to calculate the speed of any material body at the moment of interest, as well as its location.
Suppose that our Lada Vesta accelerates on the Moon, where there is no frontal air resistance due to the absence of it, then its acceleration at some stage will be stable. In this case, we will determine the speed of the car 5 seconds after the start.
V = V 0 + a ∙ t = 0 + 2.53 ∙ 5 = 12.65 m/s
or 12.62 ∙ 3600 / 1000 = 45.54 km/h
V 0 - initial speed of the point.
And at what distance from the start will our lunar vehicle be at this moment? To do this, the easiest way is to use the universal formula for determining coordinates:
x = x 0 + V 0 t + (at 2) / 2
x = 0 + 0 ∙ 5 + (2.53 ∙ 5 2) / 2 = 31.63 m
x 0 - initial coordinate of the point.
This is exactly the distance that “Vesta” will have time to move away from the starting line in 5 seconds.
But in fact, in order to find the speed and acceleration of a point in at the moment time, in reality it is necessary to take into account and calculate many other factors. Of course, if the Lada Vesta gets to the moon, it won’t be soon; its acceleration, in addition to the power of the new injection engine, is affected not only by air resistance.
At different engine speeds, it produces different forces, without taking into account the number of the engaged gear, the coefficient of adhesion of the wheels to the road, the slope of this very road, wind speed and much more.
What other accelerations are there?
Strength does more than just force the body to move forward in a straight line. For example, the gravitational force of the Earth causes the Moon to constantly bend its flight path in such a way that it always circles around us. Is there a force acting on the Moon in this case? Yes, this is the same force that was discovered by Newton with the help of an apple - the force of attraction.
And the acceleration that it gives to our natural satellite is called centripetal. How to find the acceleration of the Moon as it moves in orbit?
a c = V 2 / R = 4π 2 R / T 2, where
a c - centripetal acceleration, m/s∙s;
V is the speed of the Moon’s orbit, m/s;
R - orbital radius, m;
T is the period of revolution of the Moon around the Earth, s.
a c = 4 π 2 384 399 000 / 2360591 2 = 0.002723331 m/s∙s
Content:
Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate. Acceleration occurs only when the speed of a body changes. If the speed of a body increases or decreases by a certain constant amount, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m/s2) and is calculated from the values of two speeds and time or from the value of the force applied to the body.
Steps
1 Calculation of average acceleration at two speeds
- 1
Formula for calculating average acceleration. The average acceleration of a body is calculated from its initial and final speeds (speed is the speed of movement in a certain direction) and the time it takes the body to reach its final speed. Formula for calculating acceleration: a = Δv / Δt, where a is acceleration, Δv is the change in speed, Δt is the time required to reach the final speed.
- The units of acceleration are meters per second per second, that is, m/s 2 .
- Acceleration is a vector quantity, that is, it is given by both value and direction. The meaning is numerical characteristic acceleration, and direction is the direction in which the body moves. If the body slows down, then the acceleration will be negative.
- 2
Definition of variables. You can calculate Δv And Δt as follows: Δv = v k - v n And Δt = t to - t n, Where v to– final speed, v n– initial speed, t to– final time, t n– initial time.
- Since acceleration has a direction, always subtract the initial velocity from the final velocity; otherwise the direction of the calculated acceleration will be incorrect.
- If the initial time is not given in the problem, then it is assumed that tn = 0.
- 3
Find the acceleration using the formula. First, write the formula and the variables given to you. Formula: . Subtract the initial speed from the final speed, and then divide the result by the time interval (time change). You will get the average acceleration over a given period of time.
- If the final speed is less than the initial speed, then the acceleration has a negative value, that is, the body slows down.
- Example 1: A car accelerates from 18.5 m/s to 46.1 m/s in 2.47 s. Find the average acceleration.
- Write the formula: a = Δv / Δt = (v k - v n)/(t k - t n)
- Write the variables: v to= 46.1 m/s, v n= 18.5 m/s, t to= 2.47 s, t n= 0 s.
- Calculation: a= (46.1 - 18.5)/2.47 = 11.17 m/s 2 .
- Example 2: A motorcycle starts braking at a speed of 22.4 m/s and stops after 2.55 s. Find the average acceleration.
- Write the formula: a = Δv / Δt = (v k - v n)/(t k - t n)
- Write the variables: v to= 0 m/s, v n= 22.4 m/s, t to= 2.55 s, t n= 0 s.
- Calculation: A= (0 - 22.4)/2.55 = -8.78 m/s 2 .
2 Calculation of acceleration by force
- 1
Newton's second law. According to Newton's second law, a body will accelerate if the forces acting on it do not balance each other. This acceleration depends on the net force acting on the body. Using Newton's second law, you can find the acceleration of a body if you know its mass and the force acting on that body.
- Newton's second law is described by the formula: F res = m x a, Where F cut – resultant force, acting on the body, m– body weight, a– acceleration of the body.
- When working with this formula, use metric units, which measure mass in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m/s2).
- 2
Find the mass of the body. To do this, place the body on the scale and find its mass in grams. If you are considering a very large body, look up its mass in reference books or on the Internet. The mass of large bodies is measured in kilograms.
- To calculate acceleration using the above formula, you need to convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.
- 3
Find the net force acting on the body. The resulting force is not balanced by other forces. If two differently directed forces act on a body, and one of them is greater than the other, then the direction of the resulting force coincides with the direction of the larger force. Acceleration occurs when a force acts on a body that is not balanced by other forces and which leads to a change in the speed of the body in the direction of action of this force.
- For example, you and your brother are in a tug of war. You are pulling the rope with a force of 5 N, and your brother is pulling the rope (in the opposite direction) with a force of 7 N. The resulting force is 2 N and is directed towards your brother.
- Remember that 1 N = 1 kg∙m/s 2.
- 4
Rearrange the formula F = ma to calculate the acceleration. To do this, divide both sides of this formula by m (mass) and get: a = F/m. Thus, to find acceleration, divide the force by the mass of the accelerating body.
- Force is directly proportional to acceleration, that is, the greater the force acting on a body, the faster it accelerates.
- Mass is inversely proportional to acceleration, that is, the greater the mass of a body, the slower it accelerates.
- 5
Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resulting force acting on the body divided by its mass. Substitute the values given to you into this formula to calculate the acceleration of the body.
- For example: a force equal to 10 N acts on a body weighing 2 kg. Find the acceleration of the body.
- a = F/m = 10/2 = 5 m/s 2
3 Testing your knowledge
- 1 Direction of acceleration. The scientific concept of acceleration does not always coincide with the use of this quantity in everyday life. Remember that acceleration has a direction; acceleration is positive if it is directed upward or to the right; acceleration is negative if it is directed downward or to the left. Check your solution based on the following table:
- 2
Direction of force. Remember that acceleration is always co-directional with the force acting on the body. Some problems provide data that is intended to mislead you.
- Example: a toy boat with a mass of 10 kg is moving north with an acceleration of 2 m/s 2 . A wind blowing in a westerly direction exerts a force of 100 N on the boat. Find the acceleration of the boat in a northerly direction.
- Solution: Since the force is perpendicular to the direction of movement, it does not affect the movement in that direction. Therefore, the acceleration of the boat in the north direction will not change and will be equal to 2 m/s 2.
- 3
Resultant force. If several forces act on a body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two-dimensional space):
- Vladimir pulls (on the right) a container with a mass of 400 kg with a force of 150 N. Dmitry pushes (on the left) a container with a force of 200 N. The wind blows from right to left and acts on the container with a force of 10 N. Find the acceleration of the container.
- Solution: The conditions of this problem are designed to confuse you. It's actually very simple. Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F/m = 340/400 = 0.85 m/s 2.
Speed in physical quantity, characterizing the speed of movement and direction of movement of a material point relative to the selected reference system; by definition, equal to the derivative of the radius vector of a point with respect to time.
Speed in a broad sense is the speed of change of any quantity (not necessarily the radius vector) depending on another (more often it means changes in time, but also in space or any other). So, for example, they talk about angular velocity, rate of temperature change, speed chemical reaction, group speed, connection speed, etc. Mathematically, the “rate of change” is characterized by the derivative of the quantity under consideration.
Acceleration is denoted by the rate of change of speed, that is, the first derivative of speed with respect to time, a vector quantity showing how much the velocity vector of a body changes as it moves per unit time:
acceleration is a vector, that is, it takes into account not only the change in the magnitude of the speed (the magnitude of the vector quantity), but also the change in its direction. In particular, the acceleration of a body moving in a circle with a constant absolute velocity is not zero; the body experiences a constant magnitude (and variable in direction) acceleration directed towards the center of the circle (centripetal acceleration).
The unit of acceleration in the International System of Units (SI) is meters per second per second (m/s2, m/s2),
The derivative of acceleration with respect to time, that is, the quantity characterizing the rate of change of acceleration, is called jerk:
Where is the jerk vector.
Acceleration is a quantity that characterizes the rate of change in speed.
Average acceleration
Average acceleration> is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:
where is the acceleration vector.
The direction of the acceleration vector coincides with the direction of change in speed Δ = - 0 (here 0 is the initial speed, that is, the speed at which the body began to accelerate).
At time t1 (see Fig. 1.8) the body has speed 0. At time t2 the body has speed . According to the rule of vector subtraction, we find the vector of change in speed Δ = - 0. Then the acceleration can be determined as follows:
The SI unit of acceleration is 1 meter per second per second (or meter per second squared), that is
A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.
Instant acceleration
The instantaneous acceleration of a body (material point) at a given moment of time is a physical quantity, equal to the limit, to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:
The direction of acceleration also coincides with the direction of change in speed Δ for very small values of the time interval during which the change in speed occurs. The acceleration vector can be specified by projections onto the corresponding coordinate axes in a given reference system (projections aX, aY, aZ).
With accelerated linear motion, the speed of the body increases in absolute value, that is
and the direction of the acceleration vector coincides with the velocity vector 2.
If the speed of a body decreases in absolute value, that is
then the direction of the acceleration vector is opposite to the direction of the velocity vector 2. In other words, in this case the movement slows down, and the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.
Normal acceleration is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the vector normal acceleration perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.
In rectilinear uniformly accelerated motion the body
- moves along a conventional straight line,
- its speed gradually increases or decreases,
- over equal periods of time, the speed changes by an equal amount.
For example, a car starts moving from a state of rest along a straight road, and up to a speed of, say, 72 km/h it moves uniformly accelerated. When the set speed is reached, the car moves without changing speed, i.e. uniformly. With uniformly accelerated motion, its speed increased from 0 to 72 km/h. And let the speed increase by 3.6 km/h for every second of movement. Then the time of uniformly accelerated movement of the car will be equal to 20 seconds. Since acceleration in SI is measured in meters per second squared, acceleration of 3.6 km/h per second must be converted into the appropriate units. It will be equal to (3.6 * 1000 m) / (3600 s * 1 s) = 1 m/s 2.
Let's say that after some time of driving with constant speed the car began to slow down to stop. The movement during braking was also uniformly accelerated (over equal periods of time, the speed decreased by the same amount). In this case, the acceleration vector will be opposite to the velocity vector. We can say that the acceleration is negative.
So, if the initial speed of a body is zero, then its speed after a time of t seconds will be equal to the product of acceleration and this time:
When a body falls, the acceleration of gravity “works”, and the speed of the body at the very surface of the earth will be determined by the formula:
If you know the current speed of the body and the time it took to develop such speed from a state of rest, then you can determine the acceleration (i.e. how quickly the speed changed) by dividing the speed by the time:
However, the body could begin uniformly accelerated motion not from a state of rest, but already possessing some speed (or it was given an initial speed). Let's say you throw a stone vertically down from a tower using force. Such a body is subject to a gravitational acceleration equal to 9.8 m/s 2 . However, your strength gave the stone even more speed. Thus, the final speed (at the moment of touching the ground) will be the sum of the speed developed as a result of acceleration and the initial speed. Thus, the final speed will be found according to the formula:
However, if the stone was thrown upward. Then its initial speed is directed upward, and the acceleration of free fall is directed downward. That is, the velocity vectors are directed in opposite directions. In this case (as well as during braking), the product of acceleration and time must be subtracted from the initial speed:
From these formulas we obtain the acceleration formulas. In case of acceleration:
at = v – v 0
a = (v – v 0)/t
In case of braking:
at = v 0 – v
a = (v 0 – v)/t
In the case when a body stops with uniform acceleration, then at the moment of stopping its speed is 0. Then the formula is reduced to this form:
Knowing the initial speed of the body and the braking acceleration, the time after which the body will stop is determined:
Now let's print formulas for the path that a body travels during rectilinear uniformly accelerated motion. The graph of speed versus time for rectilinear uniform motion is a segment parallel to the time axis (usually the x axis is taken). The path is calculated as the area of the rectangle under the segment. That is, by multiplying speed by time (s = vt). With rectilinear uniformly accelerated motion, the graph is a straight line, but not parallel to the time axis. This straight line either increases in the case of acceleration or decreases in the case of braking. However, path is also defined as the area of the figure under the graph.
In rectilinear uniformly accelerated motion, this figure is a trapezoid. Its bases are a segment on the y-axis (speed) and a segment connecting the end point of the graph with its projection on the x-axis. The sides are the graph of speed versus time itself and its projection onto the x-axis (time axis). The projection onto the x-axis is not only the side side, but also the height of the trapezoid, since it is perpendicular to its bases.
As you know, the area of a trapezoid is equal to half the sum of the bases and the height. The length of the first base is equal to the initial speed (v 0), the length of the second base is equal to the final speed (v), the height is equal to time. Thus we get:
s = ½ * (v 0 + v) * t
Above was given the formula for the dependence of the final speed on the initial and acceleration (v = v 0 + at). Therefore, in the path formula we can replace v:
s = ½ * (v 0 + v 0 + at) * t = ½ * (2v 0 + at) * t = ½ * t * 2v 0 + ½ * t * at = v 0 t + 1/2at 2
So, the distance traveled is determined by the formula:
s = v 0 t + at 2 /2
(This formula can be arrived at by considering not the area of the trapezoid, but by summing the areas of the rectangle and right triangle, into which the trapezoid is divided.)
If the body begins to move uniformly accelerated from a state of rest (v 0 = 0), then the path formula simplifies to s = at 2 /2.
If the acceleration vector was opposite to the speed, then the product at 2 /2 must be subtracted. It is clear that in this case the difference between v 0 t and at 2 /2 should not become negative. When will she become equal to zero, the body will stop. A braking path will be found. Above was the formula for the time until a complete stop (t = v 0 /a). If we substitute the value t into the path formula, then the braking path is reduced to the following formula.
All tasks in which there is movement of objects, their movement or rotation, are somehow related to speed.
This term characterizes the movement of an object in space over a certain period of time - the number of units of distance per unit of time. He is a frequent “guest” of both sections of mathematics and physics. The original body can change its location both uniformly and with acceleration. In the first case, the speed value is static and does not change during movement, in the second, on the contrary, it increases or decreases.
How to find speed - uniform motion
If the speed of movement of the body remained unchanged from the beginning of the movement until the end of the path, then we are talking about movement with constant acceleration - uniform movement. It can be straight or curved. In the first case, the trajectory of the body is a straight line.
Then V=S/t, where:
- V – desired speed,
- S – distance traveled (total path),
- t – total movement time.
How to find speed - acceleration is constant
If an object was moving with acceleration, then its speed changed as it moved. In this case, the following expression will help you find the desired value:
V=V (start) + at, where:
- V (start) – the initial speed of the object,
- a – acceleration of the body,
- t – total travel time.
How to find speed - uneven motion
In this case, there is a situation where the body passed different sections of the path in different times.
S(1) – for t(1),
S(2) – for t(2), etc.
In the first section, the movement occurred at the “tempo” V(1), in the second – V(2), etc.
To find out the speed of movement of an object along the entire path (its average value), use the expression:
How to find speed - rotation of an object
In the case of rotation, we are talking about angular velocity, which determines the angle through which the element rotates per unit time. The desired value is indicated by the symbol ω (rad/s).
- ω = Δφ/Δt, where:
Δφ – angle passed (angle increment),
Δt – elapsed time (movement time – time increment).
- If the rotation is uniform, the desired value (ω) is associated with such a concept as the period of rotation - how long it will take for our object to make 1 full revolution. In this case:
ω = 2π/T, where:
π – constant ≈3.14,
T – period.
Or ω = 2πn, where:
π – constant ≈3.14,
n – circulation frequency.
- Given a known linear speed of an object for each point on the path of motion and the radius of the circle along which it moves, to find the speed ω you will need the following expression:
ω = V/R, where:
V – numerical value vector quantity (linear speed),
R is the radius of the body's trajectory.
How to find speed - moving points closer and further away
In problems of this kind, it would be appropriate to use the terms speed of approach and speed of departure.
If objects are directed towards each other, then the speed of approaching (removing) will be as follows:
V (closer) = V(1) + V(2), where V(1) and V(2) are the velocities of the corresponding objects.
If one of the bodies catches up with the other, then V (closer) = V(1) – V(2), V(1) is greater than V(2).
How to find speed - movement on a body of water
If events unfold on water, then the speed of the current (i.e., the movement of water relative to a stationary shore) is added to the object’s own speed (the movement of the body relative to the water). How are these concepts interconnected?
In the case of moving with the current, V=V(own) + V(flow).
If against the current – V=V(own) – V(current).
